Local Langlands in Families

家庭中的当地朗兰

• 批准号: 2302591
• 负责人: Gilbert Moss
• 金额: $ 26.87万
• 依托单位: University of Maine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302591&HistoricalAwards=false
• 关键词: Local; Langlands; Families

The Langlands program is a broad set of conjectures that provide an avenue toward understanding the mysterious properties of L-functions, which encode information about solutions to polynomial equations. These conjectures have motivated broad swaths of research in the field of algebraic number theory and have shed light on long-standing open problems. In some contexts, the Langlands conjectures have been proven true, but there remain subtleties that are poorly understood. In other contexts, they are yet to even be precisely formulated. This project proposes to study the Langlands program within the context of geometric deformation. In addition to the number theory research goals, the project also includes a training component for educators: a three-week summer number theory program during each year of the project for high school teachers throughout Maine, especially those serving rural areas. As part of the program, educators will develop extracurricular activities or elective courses in number theory for their local high school. The first research goal of the project is to make progress toward establishing the so-called local Langlands conjecture in families, which upgrades a presumed local Langlands correspondence to a map between moduli spaces of integral objects on either side of the correspondence, assuming a short list of desiderata. The second goal is to study the logical dependence between the local Langlands conjecture in families and the conjectures surrounding recent p-adic geometrizations of the local Langlands conjectures. The method is to generalize converse theorems and the Plancherel measure in families in order to extend a line of argument previously used in the special cases of general linear groups and classical groups. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

朗兰兹程序是一套广泛的数学模型，它提供了一条理解L函数神秘性质的途径，L函数编码了关于多项式方程解的信息。这些理论激发了代数数论领域的广泛研究，并揭示了长期存在的开放问题。在某些情况下，朗兰兹定理已被证明是正确的，但仍有一些微妙之处人们知之甚少。在其他情况下，它们甚至还没有得到精确的阐述。本项目提出在几何变形的背景下研究朗兰兹程序。除了数论的研究目标，该项目还包括一个教育工作者的培训组成部分：为期三周的夏季数论计划，在整个缅因州的高中教师，特别是那些服务于农村地区的项目每年。作为该计划的一部分，教育工作者将为当地高中开发数论方面的课外活动或选修课程。该项目的第一个研究目标是在建立所谓的局部朗兰兹猜想方面取得进展，该猜想将假定的局部朗兰兹对应关系升级为对应关系两侧的整数对象的模空间之间的映射，假设一个简短的必要列表。第二个目标是研究族中的局部Langlands猜想与围绕局部Langlands猜想的最近p-adic几何化的猜想之间的逻辑依赖性。方法是推广匡威定理和Plancherel措施在家庭中，以延长线的论点以前使用的特殊情况下，一般线性群和经典群体。该项目由代数和数论项目以及刺激竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。